| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Type I/II errors and power of test |
| Type | State probability of Type I error |
| Difficulty | Standard +0.3 This is a straightforward hypothesis testing question covering standard procedures: conducting a z-test for a mean with known variance, identifying Type I error probability (which equals the significance level), and defining Type II error in context. All parts require recall of definitions and routine application of the normal distribution test, with no novel problem-solving or complex multi-step reasoning required. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05c Significance levels: one-tail and two-tail5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Assume (pop) sd same \((0.3)\), \(H_0\): Pop mean \(= 2.4\) | B1 | |
| \(H_1\): Pop mean \(\neq 2.4\) | B1 | Allow '\(\mu\)' but not just 'mean' |
| \(\pm\frac{2.3 - 2.4}{\frac{0.3}{\sqrt{30}}}\) | M1 | Must have \(\sqrt{30}\), critical region approach \((2.293, 2.507)\) or \((2.193, 2.407)\) |
| \(= \pm 1.826\) | A1 | |
| comp \(z = \pm 1.96\) | M1 | Valid comparison (e.g. compare \(0.034\) with \(0.025\)) |
| No evidence that mean time changed | A1f | In context, allow accept \(H_0\) if correctly defined, no contradictions. One-tail test can score B1, B0, M1, A1, M1, A0 Max 4/6 |
| Total: 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(0.05\) | B1 | |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Concluding mean time has not changed when it has | B1 | OE, must have e.g. conclude/accept. SR Allow mean has decreased if a one tailed test in Part (i) |
| Total: 1 |
## Question 5(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Assume (pop) sd same $(0.3)$, $H_0$: Pop mean $= 2.4$ | B1 | |
| $H_1$: Pop mean $\neq 2.4$ | B1 | Allow '$\mu$' but not just 'mean' |
| $\pm\frac{2.3 - 2.4}{\frac{0.3}{\sqrt{30}}}$ | M1 | Must have $\sqrt{30}$, critical region approach $(2.293, 2.507)$ or $(2.193, 2.407)$ |
| $= \pm 1.826$ | A1 | |
| comp $z = \pm 1.96$ | M1 | Valid comparison (e.g. compare $0.034$ with $0.025$) |
| No evidence that mean time changed | A1f | In context, allow accept $H_0$ if correctly defined, no contradictions. One-tail test can score B1, B0, M1, A1, M1, A0 Max 4/6 |
| **Total: 6** | | |
## Question 5(ii)(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $0.05$ | B1 | |
| **Total: 1** | | |
## Question 5(ii)(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Concluding mean time has not changed when it has | B1 | OE, must have e.g. conclude/accept. SR Allow mean has decreased if a one tailed test in Part (i) |
| **Total: 1** | | |5 The time taken for a particular train journey is normally distributed. In the past, the time had mean 2.4 hours and standard deviation 0.3 hours. A new timetable is introduced and on 30 randomly chosen occasions the time for this journey is measured. The mean time for these 30 occasions is found to be 2.3 hours.\\
(i) Stating any assumption(s), test, at the $5 \%$ significance level, whether the mean time for this journey has changed.\\
(ii) A similar test at the $5 \%$ significance level was carried out using the times from another randomly chosen 30 occasions.
\begin{enumerate}[label=(\alph*)]
\item State the probability of a Type I error.
\item State what is meant by a Type II error in this context.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2018 Q5 [8]}}